def _b_power_k(self, k):
r"""
An expression involving moebius inversion in the powersum generators.
For a positive value of ``k``, this expression is
.. MATH::
\frac{1}{k} \sum_{d|k} \mu(d/k) p_d.
INPUT:
- ``k`` -- a positive integer
OUTPUT:
- an expression in the powersum basis of the symmetric functions
EXAMPLES::
sage: st = SymmetricFunctions(QQ).st()
sage: st._b_power_k(1)
p[1]
sage: st._b_power_k(2)
-1/2*p[1] + 1/2*p[2]
sage: st._b_power_k(6)
1/6*p[1] - 1/6*p[2] - 1/6*p[3] + 1/6*p[6]
"""
if k == 1:
return self._p([1])
if k > 0:
return ~k * self._p.sum(moebius(k/d)*self._p([d])
for d in divisors(k))
def primitives(n, invertible=False, q=None):
"""
Return the number of similarity classes of simple matrices
of order n with entries in a finite field of order ``q``.
This is the same as the number of irreducible polynomials
of degree d.
If ``invertible`` is ``True``, then only the number of
similarity classes of invertible matrices is returned.
.. NOTE::
All primitive classes are invertible unless ``n`` is `1`.
INPUT:
- ``n`` -- a positive integer
- ``invertible`` -- boolean; if set, only number of non-zero classes is returned
- ``q`` -- an integer or an indeterminate
OUTPUT:
- a rational function of the variable ``q``
EXAMPLES::
sage: from sage.combinat.similarity_class_type import primitives
sage: primitives(1)
q
sage: primitives(1, invertible = True)
q - 1
sage: primitives(4)
1/4*q^4 - 1/4*q^2
sage: primitives(4, invertible = True)
1/4*q^4 - 1/4*q^2
"""
if q is None:
q = QQ["q"].gen()
p = sum([moebius(n / d) * q ** d for d in divisors(n)]) / n
if invertible and n == 1:
return p - 1
else:
return p
def primitives(n, invertible=False, q=None):
"""
Return the number of similarity classes of simple matrices
of order n with entries in a finite field of order ``q``.
This is the same as the number of irreducible polynomials
of degree d.
If ``invertible`` is ``True``, then only the number of
similarity classes of invertible matrices is returned.
.. NOTE::
All primitive classes are invertible unless ``n`` is `1`.
INPUT:
- ``n`` -- a positive integer
- ``invertible`` -- boolean; if set, only number of non-zero classes is returned
- ``q`` -- an integer or an indeterminate
OUTPUT:
- a rational function of the variable ``q``
EXAMPLES::
sage: from sage.combinat.similarity_class_type import primitives
sage: primitives(1)
q
sage: primitives(1, invertible = True)
q - 1
sage: primitives(4)
1/4*q^4 - 1/4*q^2
sage: primitives(4, invertible = True)
1/4*q^4 - 1/4*q^2
"""
if q is None:
q = QQ['q'].gen()
p = sum([moebius(n / d) * q**d for d in divisors(n)]) / n
if invertible and n == 1:
return p - 1
else:
return p
